In this post, we discuss another example of proof by contradiction. We are going to prove that the solution of the equation 3^x = 8 is irrational.

Theorem

The solution to the equation is irrational.

Proof

In proof by contradiction, we assume the opposite of the theorem and then find a contradiction somewhere in the proof.

Let us assume the opposite of the theorem. We assume that the solution to the equation is rational.

Now assuming that that the solution to the equation above is rational, then is a rational number. This means that , where and are integers and not equal to (definition of rational numbers). Therefore,

Raising both sides to , we have .

Notice that this is a contradiction since is odd and is even and therefore the two numbers cannot be equal. In addition, contradicts the Unique Factorization Theorem or the Fundamental Theorem of Arithmetic.

Therefore, our assumption is false and the theorem that is irrational is true.